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hey folks boy am I tired thankfully I've
got a viewer-requested lesson for you
today so I don't have to think of
anything
Cooper 2:5 said I'm working on a math
problem and it only gave me the area of
the square no width or length and it
asks me what's the length of the
diagonal then he has a bunch of ominous
skulls and says oh okay well because you
put it in such an intimidating manner
I'm here will solve the problem let's go
through it together
remember the problem is we are given the
area of a square so let's go ahead and
start with a drawing of our square like
any good geometry problem we got our
square and we're given the area of the
square for this example let's say the
area of the squares twenty-five
twenty-five the unit's don't matter
it could be square centimeters square
inches I'm just going to write you
squared so 25 units squared whatever the
unit is doesn't matter we want to find
the length of the diagonal from the
square given the area the diagonal of
course looks something something like
that now immediately upon drawing the
diagonal just to start visualizing our
problem you should get some hint of how
we can solve this problem remember that
all of the angles in a square are right
angles so a particular interest this one
here is a right angle so by drawing this
diagonal we've just defined a right
triangle so then you might think does
Pythagorean theorem is that going to
help us out here and as it turns out it
will as it often does now the way your
your thought process might go to solve
this problem is oK we've got the area of
a square we want to find the length of
the diagonal so do we know have we
solved the similar problem with the same
unknown the unknown being the length of
the diagonal of a square and one that
might come to mind is a problem where
we're given a side length of the square
and we want to find the diagonal from
that information of course all of the
sides of a square have the same length
so if this side is s this side is also s
and look at that to find the diagonal
from this information we just use the
theorem s squared plus s squared equals
d squared but a bing bada boom there's
your answer so if we can get the side
length of the square from the area then
we're back in familiar territory a
problem that we've all probably already
done so can we get the side length from
the area of course we can because the
area of a square is just the square of
its side length so if the side length is
s the area is s squared so in particular
we have that the wall right this s
squared is equal to 25 units squared and
just take the square root of both sides
and that will give us our side length
and then we can go ahead and use the
Pythagorean theorem so we have that the
side length s is equal to the square
root of 25 units squared which is just 5
units so then we can erase these s's we
know what S is equal to now the side
lengths of our square five remember all
the side lengths are the same so they're
all five we could go ahead and throw you
in there so it's five units five units
and we just use the Pythagorean theorem
remember the Pythagorean theorem tells
us that the sum of the squares of the
legs of a right triangle is equal to the
hypotenuse squared so we've got that
five units squared plus five units
squared this is all getting squared five
units squared five units squared is
equal to the square of the hypotenuse
which is the diagonal in this case again
that's the sum of the squares of the
legs is equal to the square of the
hypotenuse five units squared is 25
units squared we already saw that up
here so 25 units squared plus 25 units
squared that's 50 units squared and so
we're almost done we've got that 50
units squared is equal to d squared just
take the square root of both sides here
and we've got that the diagonal D is
equal to the square root of 50 units
squared so we could just write that as
the square root of 50 and then the
square root of units square
is units so this is just the square root
of 50 units and that units is outside of
the square root and that is an exact
answer so if the area was 25 inches
squared
then the diagonal is root 50 inches and
we could approximate this if we wanted
to it's about seven point zero seven
units just pausing here for a moment to
point out that this same exact procedure
would give us the same exact result if
we were trying to find the other
diagonal of the square the one that goes
the other way all right on to the rest
of the lesson so I want you to try this
one on your own before watching the
solution let's say we're given that the
area of the square is 16 and we'll say
we'll work with an actual unit this time
so say 16 centimeters squared so go
ahead solve this problem just like we
did before to find the diagonal of a
square that has an area of 16
centimeters squared all right hopefully
you've got an answer now when I go
through this problem I'm going to leave
out the units once you do a couple of
these problems you know that if you're
given an area in centimeters squared for
example your side length is going to be
in centimeters if you're given your area
in inches squared your side length is
going to be in inches and so on so we
can leave off the units we know what the
units will be at the end so how do we
solve this problem same way we did
before the area is 16 centimeters
squared which is the square of the side
length of the square so then we just
take the square root of both sides
square root of that square root of that
so leaving off our units we have that
the side length of the square is equal
to the square root of 16 what's the
square root of 16 well of course that's
4 so then we can go ahead and erase our
s's over here we know that the side
length of this square are all four now
we've got a right triangle with leg
lengths of four we just go ahead and
apply our favorite Pythagorean theorem
that tells us that the sum of the
squares of the leg lengths four squared
plus four squared that's the sum of the
squares of the
leg lengths is equal to the diagonal
squared d squared the hypotenuse which
in this case is our diagonal that we're
looking for 4 squared plus 4 squared
that's 16 plus 16 and that is 32 so 4
squared plus 4 squared 16 plus 16 that's
32 is equal to the diagonal squared take
the square root of both sides and we've
got our answer if you liked the video of
the square is equal to the square root
of 32 so since our area was given in
centimeters squared
we know that this diagonal is the square
root of 32 centimeters long and if you
approximated this square root you would
have got about five point six six
centimeters and that's all there is to
it given the area of a square the square
root of that is the side length of the
square those are our legs of this right
triangle and then we just apply the
Pythagorean theorem side length squared
plus side length squared is equal to the
hypotenuse squared and in this case the
hypotenuse is our diagonal and of course
the answer falls out immediately from
that if you haven't seen a proof of the
Pythagorean theorem by any chance I'll
leave a link in the description to a
lesson where I prove the Pythagorean
theorem and even if you're not big on
the geometry and the proofs I'm eating
bugs the whole time so you might find it
a little bit more amusing in any event I
hope this video helped you understand
how to find the length of a diagonal
from the square when given the area be
sure to let me know in the comments if
you have any questions need anything
clarified or have any other video
requests thank you very much for
watching I will see you next time and be
sure to subscribe for the swankiest math
lessons on the Internet
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